I realise the irony of me taking the piss out of the many bad Blender tutorials out there...then posting my own Blender tutorial. But this problem; mesh-to-rig parenting failure - had me stumped for days before I finally found the solution.
Position the bones of your rig to the appropriate joints on your character's mesh. I used the Rigify Blender add-on and chose the human meta-rig.
With that done, go into object mode and toggle X-ray view on (so you can see your rig inside your model).
Select the model mesh (this should be one single, joined solid object - not several different parts)
Shift + select your rig.
Ctrl+P to parent mesh to the rig
Set parent to 'With Automatic Weights'.
Then go into Pose mode.
Select a bone. If it doesn't move or rotate the mesh when you press G or R*, then the parenting has failed.
A MAJOR REASON WHY PARENTING MESH TO RIG FAILS IS BECAUSE YOUR MESH IS TOO HIGH-POLY!
You will need to take a duplicate of your mesh (preserve the high-poly original), switch to edit mode and then select the Mesh menu, scroll down to Clean Up, and select Decimate Geometry. I tested it by going straight for a Ratio of 0.5 (halving the number of polys I guess?), and then when I parented this simpler mesh to the rig, it finally fucking moved properly!
The weights are shit and the mesh looks horrendous, but next steps are fine-tuning to see how many polys I can preserve before rigging failure occurs, and improving the weighting of individual bones.
* It helps to have the 'show gizmo' button near the top right corner of the 3D view (the curve with an arrow pointing through it) set so that all 'object gizmos' are showing; especially move and rotate. This way the x,y,z movement circles will appear when you select a bone.